Square roots and Heron’s method
Step by step, a rectangle turns into a square with the same area. The method is almost 2000 years old and reaches √2 surprisingly quickly.
Mathematics – grade 9
Unit 1Square roots
Definition of square roots, irrationality and real numbers, Heron’s method, calculating with roots.
Unit 2Quadratic functions
2.1Quadratic functions and quadratic equations
Effect of a, d and e on the parabola, finding the vertex, reading off properties, the quadratic formula; standard, vertex and factored form.
Quadratic function
Stretch, reflect and shift parabolas and read off the vertex and the zeros.
Solving equations graphically: intersections of graphs
Draw both sides of an equation as graphs. The solutions are where the graphs cross.
2.2Quadratic functions in applications
Systems with three unknowns (parabola through three points), intersecting lines and hyperbolas, optimisation problems.
Parabola through three points
Drag three points and the matching parabola always passes through all of them. The equation is worked out step by step.
Optimisation problems with parabolas
Which enclosure has the largest area? Picture and graph show the maximum, first by trying, then by exact calculation.
Unit 3Probability of combined events
Intersection and union of events, Venn diagrams and two-way tables, the addition rule.
Two-way tables and Venn diagrams
A two-way table, a Venn diagram and a tree diagram show the same data and are linked. Change one and the others follow.
Unit 4Similarity and the intercept theorem
Similar figures, the intercept theorems and how area and volume change when scaling.
Intercept theorems
How tall is the tree without climbing it? Two rays and a parallel line are enough, because the ratios stay the same.
Similar figures: how do area and volume grow?
Enlarge lengths, areas and solids. Doubling the length makes the area four times and the volume eight times as large.
Unit 5Power functions with natural exponents and extending powers
Graphs of y = a · xⁿ and their symmetry, nth roots, powers with rational exponents.
Power functions y = a · xⁿ
Even or odd, positive or negative: y = a · xⁿ changes with every exponent, and only a few points always stay the same.
Unit 6Pythagorean theorem
Proof, converse and many applications in right triangles.
Pythagorean theorem
Squares on the sides of a right triangle and a puzzle proof show: a² + b² = c².
Unit 7Trigonometry
7.1Trigonometry in right triangles
Ratios of sides as sine, cosine and tangent, relationships between them and calculations in applications.
Sine, cosine and tangent in right triangles
In a right triangle, the ratio of two sides depends only on the angle. This gives sine, cosine and tangent, with examples such as ramps and ladders.
7.2Law of sines and law of cosines
Sine and cosine for angles up to 360° on the unit circle, the laws of sines and cosines.
Unit circle
Sine, cosine and tangent on the unit circle, and how the sine curve grows out of it.
Law of sines and law of cosines
Calculate sides and angles in any triangle with the law of sines and the law of cosines. A helper shows which law fits your problem.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): LehrplanPLUS, Gymnasium, Mathematics, grade 9